Math 119 A,B, Syllabus


Ordinary Differential Equations, Chaotic Dynamics and Bifurcation Theory

• Linear Equations
  • First order systems of ODEs
  • Solution of ODEs by diagonalization
  • Operators and exponentials of matricies
  • Linear two by two systems
  • Higher order systems
• Existence, Uniqueness and Invariant Manifolds
  • The Picard existence theorem
  • Uniqueness
  • Examples of non-uniqueness and blow-up
  • The nonlinear pendulum and Duffing's equation
  • The invariant manifold theorem
• The Geometry of Phase Space
  • Orbits and diffeomorphism
  • Vector fields and flows
  • Flow equivalence
  • The rectification theorem
  • Continous dependence on initial data
  • Topological conjugacy
  • Classification of flows
• Stability
  • Global solutions
  • Lyapunov functions
  • The Lorentz equations
  • $\alpha$ and $\omega$ limit sets
  • Absorbing sets and attractors
  • Basic attrators
  • The Lorentz attractor
• Chaos
  • The Poincaré map
  • The Smale Horseshoe
  • Symbolic dynamics
  • Strange hyperbolic sets
  • The Melnikov method
• Hamiltonian Systems
  • Liouville's theorem
  • The Kolmogorov-Arnold-Moser theorem
  • Circle maps
• Center Manifolds and Bifurcation Theory
  • The center manifold theorem
  • The Lorentz center manifold
  • Bifurcation theory
  • Codimension one bifurcations
    • The saddle node bifurcation
    • The transcritical bifurcation
    • The pitchfork bifurcation
  • Codimension two bifurcations
    • The Hopf bifurcation
  • Normal forms
• Maps and Diffeomorphisms
  • One-dimensional maps
  • The logistic map
  • The flip bifurcation
• The Feigenbaum Period-Doubling Cascade
  • The quadradic map
  • The Feigenbaum conjectures
  • The characterization of the strange attractors